9 MEASURABLE FUNCTIONS 143 directly (13.5.5) that, for any mapping g : X -* R, we have fi*(g) = v*(g \ Y), and hence that such a mapping g is /j-integrable if and only if g \ Y is v- integrable, in which case we have ftolYX.-f, J J (13.9.20.2) Likewise, a mapping u of X into a topological space Z is //-measurable if and only if u \ Y is v-measurable. (13.9.21) If n : X -+ X' is a homeomorphism, then a mapping u' of X' into a topological space Y is 7i(/i)-measurable if and only if u o n is ^-measurable. PROBLEMS 1. Let (fmn) be a double sequence of /x-measurable mappings of X into a metric space Y. Suppose that for each m the sequence (fmn)n-*i converges almost everywhere to a function gm, and that the sequence (gm) converges almost everywhere to a function h. Show that there exist two strictly increasing sequences of integers (mk), (nk) such that the sequence (fmknk)kzi converges almost everywhere to h. (First prove the result when X is compact by observing that, for each e> 0, there exists a compact subset K c X such that jit(X — K) <> e and such that the sequence (gm) and each of the sequences (/mn)n=si are uniformly convergent on K. Then pass to the general case by using the diagonal trick.) 2. (a) Let/, g be two real-valued functions ^0, and suppose that g is /z-measurable. Show that ju,*0^) = inf ^(ug), where u runs through the set of all /^-measurable functions ^/. (b) Let/, g, h be three functions ^0 on X, finite or not, and suppose that g and h are /^-measurable. Show that p*(f(g 4- h)) = fi*(fy) -f ft*(//z) (using the convention of (13.11) for the product of 0 and + °o). 3. Let/: X -> R be /x-measurable. For each n E Z, let An be the set of all x e X such that 2"-1 < |/(*)| g 2". Show that/is integrable if and only if 2 2>(An) < + oo. n 6 Z 4. Let (/„) be a sequence of integrable functions on X, converging simply to a function/, (a) If/is integrable and \ fdp.= lim /„ dp, show that for each e > 0 there exists J JT-+-00 J an integrable set A, an integrable function g J> 0 and an integer «0 such that, for all «J>/z0, we have fnd^ <Je and \fn(x)\ g \g(x)\ for all xeA. (Consider an integrable set B such that |/| dp <£!« and such that/is bounded on B, and apply J X— B EgorofT's theorem.)ible set N, such that u \ Kn is continuous for all n ; hence (c)